Small-time asymptotics of heat kernels of one-dimensional diffusions in a random environment
Abstract
We establish the Varadhan small-time asymptotics for the quenched and annealed heat kernels of one-dimensional diffusions in a random environment with generator ℒW f(x)=e−ρ(x,W)(ea(x,W)f'(x))'. The coefficients a and ρ are continuous in space and satisfy a local exponential moment condition. We assume that the law of the intrinsic coordinate map ΛW has compact support ℒ consisting of strictly increasing functions. Let qW(t,x,y) and q(t,x,y)=𝔼[qW(t,x,y)] denote the quenched and annealed heat kernels with respect to Lebesgue measure, respectively. We prove that, for almost every environment W, limt↓0tlog qW(t,x,y)=−1/2|ΛW(y)−ΛW(x)|2, and that limt↓0tlog q(t,x,y)=−1/2minΛ∈ℒ|Λ(y)−Λ(x)|2. Both limits hold uniformly on compact subsets of ℝ2. The framework includes Brox diffusion, formally described by dXt=dBt−1/2Ẇ(Xt) dt, where B is a standard Brownian motion and W is an independent two-sided Brownian motion.
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BibTeX
@misc{wang2026smalltime,
title = {{Small-time asymptotics of heat kernels of one-dimensional diffusions in a random environment}},
author = {Yiduo Wang and Saisai Yang and Tusheng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21345},
eprint = {2609.21345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21345},
}
RIS
TY - UNPB
TI - Small-time asymptotics of heat kernels of one-dimensional diffusions in a random environment
AU - Yiduo Wang
AU - Saisai Yang
AU - Tusheng Zhang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21345
UR - https://arxiv.org/abs/2609.21345
AB - We establish the Varadhan small-time asymptotics for the quenched and annealed heat kernels of one-dimensional diffusions in a random environment with generator ℒ_W f(x)=e^−ρ(x,W)(e^a(x,W)f'(x))'. The coefficients a and ρ are continuous in space and satisfy a local exponential moment condition. We assume that the law of the intrinsic coordinate map Λ_W has compact support ℒ consisting of strictly increasing functions. Let q^W(t,x,y) and q(t,x,y)=𝔼[q^W(t,x,y)] denote the quenched and annealed heat kernels with respect to Lebesgue measure, respectively. We prove that, for almost every environment W, lim_t↓0tlog q^W(t,x,y)=−1/2|Λ_W(y)−Λ_W(x)|^2, and that lim_t↓0tlog q(t,x,y)=−1/2min_Λ∈ℒ|Λ(y)−Λ(x)|^2. Both limits hold uniformly on compact subsets of ℝ^2. The framework includes Brox diffusion, formally described by dX_t=dB_t−1/2Ẇ(X_t) dt, where B is a standard Brownian motion and W is an independent two-sided Brownian motion.
ER -